Timing of Network Synchronization By Refractory Mechanisms

Urs Achim Wiedemann, Anita Lüthi


Even without active pacemaker mechanisms, temporally patterned synchronization of neural network activity can emerge spontaneously and is involved in neural development and information processing. Generation of spontaneous synchronization is thought to arise as an alternating sequence between a state of elevated excitation followed by a period of quiescence associated with neuronal and/or synaptic refractoriness. However, the cellular factors controlling recruitment and timing of synchronized events have remained difficult to specify, although the specific temporal pattern of spontaneous rhythmogenesis determines its impact on developmental processes. We studied spontaneous synchronization in a model of 600–1,000 integrate-and-fire neurons interconnected with a probability of 5–30%. One-third of neurons generated spontaneous discharges and provided a background of intrinsic activity to the network. The heterogeneity and random coupling of these neurons maintained this background activity asynchronous. Refractoriness was modeled either by use-dependent synaptic depression or by cellular afterhyperpolarization. In both cases, the recruitment of neurons into spontaneous synchronized discharges was determined by the interplay of refractory mechanisms with stochastic fluctuations in background activity. Subgroups of easily recruitable neurons served as amplifiers of these fluctuations, thereby initiating a cascade-like recruitment of neurons (“avalanche effect”). In contrast, timing depended on the precise implementation of neuronal refractoriness and synaptic connectivity. With synaptic depression, neuronal synchronization always occurred stochastically, whereas with cellular afterhyperpolarization, stochastic turned into periodic behavior with increasing synaptic strength. These results associate the type of refractory mechanism with the temporal statistics and the mechanism of synchronization, thereby providing a framework for differentiating between cellular mechanisms of spontaneous rhythmogenesis.


Essential functions of the CNS, such as locomotion, arousal, and sensory processing, are associated with the appearance of synchronized rhythmic electrical activity (Delcomyn 1980; Engel et al. 2001; Marder and Calabrese 1996; Steriade et al. 1993a). For the autonomous generation of many of these activities, specialized pacemaker circuits are crucial (Lüthi and McCormick 1998; Marder and Calabrese 1996; Steriade and Deschênes 1984). On the other hand, even in the absence of built-in pacemakers, episodic activity can arise spontaneously in neuronal networks (Feller 1999; Katz and Shatz 1996; O'Donovan 1999; Spitzer 2002).

Such spontaneous neuronal synchronization is a characteristic feature of immature networks (Feller 1999; O'Donovan 1999; Zhang and Poo 2001) but is also found in adult neocortex (Sanchez-Vives and McCormick 2000). During prenatal periods, in which a major source of excitation via sensory inputs is lacking, spontaneous synchronization is widely regarded to play a fundamental role in the establishment of a functional network architecture, such as in the visual system, in the spinal chord, and in the hippocampus (Ben-Ari 2001; Feller 1999; Katz and Shatz 1996; O'Donovan 1999). The origin of this autonomous organization of neural networks has been generally associated with three major characteristics of developing networks (Ben-Ari 2001; Feller 1999; O'Donovan 1999): 1) comparatively high and homogeneous synaptic interconnectivity, 2) a predominance of excitatory over inhibitory synaptic interactions, and 3) expression of a transient negative feedback that reduces the excitability of the network following an episode of synchronization. Such refractory processes can involve intrinsic or synaptic properties of individual neurons and have been experimentally identified as a change in membrane potential [e.g., afterhyperpolarization (AHP); Darbon et al. 2002; Sanchez-Vives and McCormick 2000] or as a decrease in synaptic connection strength (e.g., short-term synaptic depression; Fedirchuk et al. 1999; Harris et al. 2002; Tabak et al. 2001). These characteristics suggest a qualitative picture of rhythmogenesis in which the timing of synchronized discharges is determined by the time constant of recovery from neuronal or synaptic refractoriness, thus plausibly rendering an active pacemaker mechanism obsolete.

This picture is also supported by model studies, which follow the time course of average network properties, such as average discharge rates and overall refractoriness (Senn et al. 1996; Tabak et al. 2000). Moreover, computational models of retina and cortex emphasize the importance of spontaneous action potential discharge (Butts et al. 1999; Timofeev et al. 2000) and of refractory processes (Tsodyks et al. 2000) in the initiation and timing of synchronization, respectively. However, these models do not address how distinct types of refractoriness control neuronal recruitment and temporal statistics during spontaneous synchronization. Both recruitment and timing, however, are regarded as crucial, in particular, in developing networks, for establishing functional connectivity patterns, activation of enzymes, and expression of genes (De Koninck and Schulman 1998; Berridge et al. 2000; Stellwagen and Shatz 2002).

Here we show that two diverse refractory mechanisms, in combination with random fluctuations intrinsic to the network, lead to spontaneous synchronization with distinct temporal statistics. We explore this idea in a model of leaky integrate- and-fire neurons, which is known to show spontaneous synchronization, as pioneered by Tsodyks et al. (1998, 2000).


We simulated recurrent networks of 600–1,000 excitatory and ≤150 inhibitory leaky integrate-and-fire neurons that were randomly interconnected with a pairwise contact probability of 5–30%. The membrane potential Vi of the ith neuron was evolved according to Math(1) with τ = 30 ms. After reaching firing threshold Vthr = –45 mV, the membrane potential was reset to –55 mV for 3 ms.

The time-dependence of the ith synaptic current was determined by Math(2) Here, the maximal connection strength Aij between neurons i and j was set to the nonzero “synaptic connection strength” A with a 5–30% probability. The time dependence of the active synaptic resources yij(t) is given by Math(3) with tI = 3 ms. The value xij(t) represents the synaptic reserve from which a fraction u = 0.5 is drawn upon generation of an action potential at time tsp(j).

Two different mechanisms of refractoriness were studied.

Model 1 “with synaptic depression”

Following Tsodyks et al. (1998, 2000), the use-dependence of synaptic resources was modeled by a system of kinetic equations Math(4) Math(5) with τrec = 800 ms. Here xij, yij, and zij are the fractions of synaptic resources in the recovered, active, and inactive state of the synapse ij. For the nonspecific background currents Ib(i) in Eq. 1, potentials RinIb(i) were spread according to a flat distribution of Vspread = 4 mV centered such that one-third of the population was capable of spontaneous firing without synaptic input.

Model 2 “with AHP”

The postsynaptic current was modeled without use-dependent synaptic modification, i.e., xij(t) = 1.0 = constant in Eq. 3. Refractoriness was implemented by an AHP in Eq. 1, which decreased the membrane potential over a timescale τAHP = 600 ms after spiking Math(6)

This term was subtracted from the right-hand side of Eq. 1, and the membrane potential Vi of the ith neuron was evolved according to Math(7)

The model with AHP was fully specified by determining the background currents Ib(i) and the amplitude of AHP, vAHP(i). In test runs, we tuned the parametrizations of RinIb(i) and vAHP(i) such that the distribution of generated average firing frequencies (see Fig. 7B) was comparable to that in the model with synaptic depression (see Fig. 7A). All results presented in this work were obtained for a sigmoidal distribution of vAHP(i) among the N excitatory neurons in the network, vAHP(i) = wAHP {1 + exp[(inth)/8]}–1, 0 < i < N, where wAHP = 3 mV and nth = 0.67N. The background currents Ib(i) were chosen as RinIb(i) = Vthr + (Vspread/N)(inth) for i > nth and RinIb(i) = Vthr + (Vspread/20N)(inth) for i < nth. Since it is the combination RinIb(i) – VAHP(i,t) that drives the membrane potential in Eq. 7, the reduced spread of RinIb(i) for i < nth can be viewed as balancing partly the large hyperpolarization term vAHP(i) for these cells. Thus in the absence of external input, two-thirds of the excitatory neurons were not capable of firing but moved close to firing threshold when the AHP current decayed.

fig. 7.

Statistics of PS generation are independent of synaptic strength in the model including synaptic depression but not in the one including cellular AHP. A and B, left: time-triggered averages of neuronal discharges of the 2 models for high A (4.0 for the model with synaptic depression, 0.7 for the model with AHP). Color code is the same as the one used in Fig. 4. Right: distribution of intrinsic firing frequencies of neurons in the 2 models. C: time course of PS generation for 2 different synaptic strengths in the model with synaptic depression. Although the frequency of PSs is increased with greater synaptic strength, irregularities in interspike intervals remain. D: time course of PS generation for 2 different synaptic strengths in the model with cellular AHP. With greater synaptic strength, interspike intervals become highly regular, and the deviation from the average interspike intervals diminishes strongly. For C and D, PSs are illustrated schematically with vertical lines. The graphs illustrate •, average interspike interval; ○, normalized deviation from average interspike intervals as a function of the synaptic connection strength A.

For both the model with synaptic depression and the model with AHP, our standard simulation implemented a network of N = 600 excitatory cells connected with a pairwise probability of 10% and with no inhibitory cells included in the simulation. Except where explicitly stated otherwise, the results presented below are obtained for this standard setting. In particular, all figures show simulation results for purely excitatory networks.

χ2 values quoted for the distributions in Figs. 3 and 5 were determined by associating statistical errors only to the histograms of interspike intervals. Thus χ2 values of order 1 (much greater than 1) quantify the agreement (systematical deviations) between the simulated interspike interval distributions and the conjectured distribution (Eq. 8).

fig. 3.

Generation of PSs is paralleled by a marked decrease of synaptic resources in the model with synaptic depression. A, top: spontaneous generation of PSs for a period of 30 s, A = 2.6 mV. Bottom: associated change in the total synaptic reserve [equal to the sum over xij(t) for all synapses] and its gradual recovery during periods free of PSs. Dotted lines denote the temporal coincidence of a PS with the marked decrease in the synaptic reserve. B: monoexponential fit of expanded portions of trace in A. C: distribution of interspike intervals for 357 PSs from a simulation including the portion presented in A. Thick line indicates the fit to Eq. 8. χ2/ndf = 1.4.

fig. 5.

In the model with AHP, the generation of PSs is paralleled by a marked increase in the net AHP amplitude. A, top: spontaneous generation of PSs for a period of 30 s, A = 0.2 mV. These values were approximately 10-fold larger for the model with synaptic depression compared with that with AHP, consistent with average (xij) « 1 for the former, and average (xij) = 1 for the latter model. Bottom: associated change in the total AHP (equal to the sum of AHP amplitudes Formula for all neurons) and its recovery during periods free of PSs. This recovery deviates from a monoexponential function and shows fluctuations reflecting intermittent firing of neurons. Dotted lines denote the temporal coincidence of a PS with the change in Formula. B: monoexponential fit of expanded portions of trace in A, revealing deviations from a monoexponential recovery at large times. C: distribution of interspike intervals for 226 PSs from a simulation lasting 731 s including the portion presented in A. Thick line indicates the fit according to Eq. 8. χ2/ndf = 10.8.


Characteristics of the network model

We studied the long-term behavior (≤1,000 s) of recurrently connected networks of 600–1,000 excitatory and ≤150 inhibitory integrate-and-fire neurons (Eq. 1), which were randomly interconnected with a probability of 5–30% via synapses that were implemented according to Eq. 2. To provide the network with a basal level of intrinsic activity, one-third of all neurons were equipped with a background current Ib that depolarizes the membrane potential above the threshold voltage Vthr = –45 mV (Eq. 1). The remaining two-thirds of neurons had a value of Ib that results in membrane voltages smaller than Vthr, preventing neurons from discharging in the absence of suprathreshold excitatory input. This distribution of currents results in a network of neurons, one-third of which exhibited on-going action potential discharge (Fig. 1, A and B). The spread of intrinsic properties of these neurons and their random connectivity resulted in a flat background of spontaneous asynchronous activity, despite the absence of an external noise source.

fig. 1.

Population spikes (PSs) similar in size and duration are generated spontaneously in recurrent models with either synaptic depression (left) or neuronal afterhyperpolarization (AHP; right). A and B: typical firing patterns of the two networks. Graphs show the number of neurons discharging per millisecond. C and D: expanded portions of A and B illustrating the time course of a PS in more detail. E and F: time plot of neuronal discharges (each marked by a cross) for each individual neuron in the network, numbered from 1 to 600. Note the recruitment of most neurons during a PS. In F, a higher portion of neurons discharges accidentally during periods free of PSs (see results).

Synaptic refractoriness was modeled by implementing a use-dependent synaptic depression formalism described previously (Eqs. 3–5; Tsodyks et al. 1998, 2000). To model cell-intrinsic refractoriness, each single neuron was equipped with a slowly decaying intrinsic membrane AHP VAHP(i,t) with a decay time constant of 600 ms (Eq. 6) that was activated following action potential discharge.

Spontaneous occurrence of population spikes

Figure 1 shows the spontaneous steady-state behavior of the network during a period of 8 s for both synaptic depression (Fig. 1A) or intrinsic AHP (Fig. 1B) dominating network refractoriness. Both networks show an on-going basal, apparently asynchronous, discharge of action potentials (approximately 5 active cells/ms). At recurring intervals, however, the number of cells discharging an action potential increased abruptly and peaked at a factor 10–15 above background activity, indicating the occurrence of simultaneous action potential discharge of a large fraction of neurons in population spikes (PSs) (Tsodyks et al. 1998, 2000). Figure 1, C and D, shows that the generation of a PS occurs in a narrow time window of ±20 ms, during which >80% of all neurons are recruited. This resets the membrane potentials of all participating neurons almost simultaneously, leading to a reduction of firing activity from which spontaneously active neurons recover on a time scale set by their intrinsic membrane time constant (30 ms). Inspection of single-cell voltage traces (data not shown) confirms that this leads to a full recovery of the basal activity within <50 ms. Comparable characteristics of basal firing rates (approximately 5 active cells/ms), peak heights of PSs (factor 10–15 above baseline), and recruitment of neurons within a PS (>80%) were found for both the model with intrinsic AHP and the model with synaptic depression. Therefore the spontaneous generation of PSs per se does not depend on the mechanistic basis of slow refractory processes within the network. Indeed, comparing the discharge time of each individual neuron relative to the PS, we found that, in both models, a PS results from the rapid recruitment of the neuronal subpopulation that is otherwise predominantly silent (Fig. 1, E and F). Thus spontaneous synchronization of neurons within neuronal networks can occur independently of the precise implementation of use-dependent mechanisms of refractoriness and is therefore not specific to these. In agreement with earlier findings (Tsodyks et al. 2000), inclusion of inhibitory neurons did not affect this network behavior, provided the strength of inhibition was not too great (data not shown). For the further analysis of the statistics of PS generation, the majority of simulations were therefore run without inhibitory neurons.

Generation of PSs is observed over a wide range of model parameters

Figure 2 makes it clear that the generation of PSs was observed over a wide range of model parameters, including sufficiently large connection strength and connectivity. Similar dependencies on these parameters were observed for N = 600, 800, and 1,000 (data not shown). Moreover, for fixed connection probability, comparable average numbers of PSs were observed for networks of N = 600, 800, and 1,000 neurons after readjusting the connection strength A proportional to 1/N2. This rescaling of A is consistent with the expectation that it is the average excitatory input per cell that determines firing. Thus spontaneous synchronization was a phenomenon that persisted over a large volume of parameter space, indicating that it was a canonical property of recurrently connected networks.

fig. 2.

Behavior of the model over parameter space for N = 1,000. The average number of PSs generated per 50 s is shown as a function of the connection probability between neurons and of the connection strength A: model with synaptic depression. B: model with AHP. Both models reveal a sharp onset of PSs above critical values for both parameters. To clearly present this onset, values of PSs/50 s above 100 were truncated in A.

Initiation and timing of PSs in the model with synaptic depression

Both synaptic depression and cellular AHP have been proposed to underlie the refractory period in spontaneously synchronizing networks in vitro (Darbon et al. 2002; Fedirchuk et al. 1999; Harris et al. 2002; Sanchez-Vives and McCormick 2000). To determine how synchrony arises dynamically, we aimed at correlating the degree of refractoriness in the network with the occurrence of PSs. To this end, we quantified the degree of synaptic depression by the total available synaptic reserve in the network [equal to the sum of the available synaptic resources xij(t) for all synapses, xtot(t)], calculated for each time step in the simulation.

Figure 3A (top) shows a sequence of 30 s during which 11 PSs occur. The total synaptic reserve decreases dramatically during a PS and slowly recovers within the subsequent time period devoid of a PS (bottom). The time course of recovery was well fitted by a monoexponential curve, and a sample of 10 fitted traces yielded time constants between 700 and 800 ms (2 examples shown in Fig. 3B). The fitted time constants are close to the value of τrec = 800 ms implemented in the network (Fig. 3B), consistent with the predominant involvement of recovery from synaptic depression in the recuperation of the total synaptic reserve.

Closer inspection of the temporal alignment of the occurrence of PSs with the time course of the total synaptic reserve shows, however, that the PS occurring shortly before the 15th second in Fig. 3A is followed by a prolonged silent period during which xtot recovers to its maximal level, but the network fails to generate another PS for approximately 5 s. In contrast, during the interval of equal length between 4 and 9 s, three PSs are generated, even though xtot has not recovered fully. Thus interspike intervals spread over a considerable time between 0.5 and >6 s and their distribution peaks around 1.5 s. This is seen from the histogram of the distribution of interspike intervals (Fig. 3C), which resulted from the evaluation of 357 PSs generated from a simulation lasting 783 s. In conclusion, although synaptic depression strongly increases during a PS and reduces the probability of occurrence of another PS in the following time window (see next paragraph), the next PS cannot be predicted deterministically from inspecting the average refractoriness of the network.

We therefore tested the assumption that the timing of PSs is determined probabilistically as a product of two processes: 1) for maximally recovered xtot, occurrence of PSs becomes stochastic and is independent of preceding PSs; and 2) for incompletely recovered xtot, the probability of PSs is reduced due to decreased synaptic transmission. Assumption 1) implies that the temporal distribution of PSs follows Poisson statistics and that interspike intervals are hence distributed monoexponentially with a time constant τpoisson. According to assumption 2), this monoexponential behavior is modified by a recovery function, which depends only on the time since the last PS. The recovery function was modeled by a sigmoidal curve, which suppresses the occurrence of PSs for interspike intervals smaller than τref and switches on over a time scale τspread < τref. This yields a conjectured distribution of interspike intervals of the form Math(8) This distribution was fitted to the model data (thick line in Fig. 3C), yielding a refractory offset τref = 1,195 ± 18 ms with spread τspread = 96 ± 12 ms. The time constant τpoisson = 1,059 ± 53 ms characterizes the Poisson distribution for fully recovered xtot. The absolute size of this distribution is determined by the prefactor n, which is proportional to the number of PSs in the sample (for the sample chosen here, n = 251 ± 25). Based on the good agreement between the conjectured distribution and the model data (χ2/ndf = 1.4), we conclude that the timing of PSs in a model with synaptic depression is essentially stochastic.

The apparently asynchronous background activity of spontaneously active neurons is the predominant source of stochastic behavior, suggesting that random fluctuations in background activity may serve as triggers for PSs. To test this hypothesis, we grouped neurons according to their spontaneous activity. As evident from Fig. 1E, the “fast cells,” represented by neurons 390–600, provide the flat background activity. The “middle cells” (neurons 320–389) show little spontaneous activity but are relatively easily excitable in comparison to the “slow cells,” which comprise the predominantly silent neurons (neurons 1–319). Further motivation for this classification came from a study of Tsodyks et al. (2000), in which it was recognized that elimination of a subgroup of moderately active neurons prevents generation of PSs.

Time-triggered averages of PSs reveal that the three neuronal subgroups enter the synchronization process in a specific temporal sequence, which can already be seen from inspecting single PSs in Fig. 4A. Thus the constant background activity at 100 ms before a PS consists mainly of the on-going discharge of the fast group (note overlay of the lines “all cells” and “fast cells” in Fig. 4, A and B), whereas the majority of middle and slow cells are silent. Thirty to 50 ms before the occurrence of a PS, spontaneous discharges start to arise in the middle group (red lines in Fig. 4, A and B) and reach a peak approximately 10 ms before the maximum of the PS. This background-induced fluctuation in the activity of the middle cells triggers the immediate upstroke of the discharge of the slow cells. The peak of the activity of the group of slow cells then contributes the major part of the peak of the PS (76.3% in Fig. 4B), while the fast cells generate a peak that is delayed by 2.2 ms (Fig. 4B, bottom). The second peak of the middle cells is due to neurons that fired in the initiation phase of the PS and thus recovered sufficiently (within approximately 30 ms) for being recruited again after the maximum of the PS. Except for these middle cells and some fast cells, almost all neurons fired exactly once per PS and only a small fraction (<20%) remained silent.

fig. 4.

Generation of PSs occurs via a temporally defined recruitment of three neuronal subpopulations. A: two examples illustrating that slow, middle, and fast cells discharge in a specific temporal order within a PS. Color-coded lines depict the time course of activity of fast cells (green lines), middle cells (red lines) and slow cells (blue dotted lines). B, top: PS-triggered averages for these three neuronal populations for ±100 ms around the peak of the PS (averaged over 357 PSs). Bottom: fraction of neurons from the three neuronal subpopulations recruited during a PS, highlighting that the discharge of the middle cells precedes the recruitment of the other neuronal subgroups. C: representative membrane potential traces from neurons from the slow (cells 1, 100, 200, 300) and the middle group (cell 350) during the generation of a PS. In the process of synchronization, neurons further away from threshold receive progressively amplified excitatory postsynaptic potentials (EPSPs) arriving within a short time period (<10 ms; arrows). For clearer presentation of these EPSPs, the traces are displaced horizontally by 100 ms.

This sequential, avalanche-like recruitment of neuronal subgroups during a PS did not depend on the exact distribution of neurons within the groups as long as the ordering of neurons according to their spontaneous activity was preserved (data not shown). Moreover, during periods free of PSs, the summed discharges of the middle group never yielded any significant fluctuations in their activity above average, indicating that the fluctuations in the time-triggered averages are connected to the ensuing generation of a PS.

Figure 4C shows that moderately active neurons from the middle group are essential for the initial amplification mechanism leading to PSs: a typical middle cell (cell 350) receives excitatory input from the background of spontaneously active fast cells, resulting in fluctuations that are typically subthreshold but can cross threshold without a qualitative change in synaptic synchronization. However, due to their relatively strong synaptic strength, middle cells increase the synchronized synaptic input for neurons further away from threshold, thus serving as sensitive amplifiers for fluctuations in the background activity. This is clearly seen by inspecting the size of synaptic inputs in neurons representing the slow group (Fig. 4C). The progressive marked increase in amplitude of excitatory postsynaptic potentials (EPSPs) received by neurons further away from threshold shows that increased synchronization of presynaptic inputs is necessary to bring these silent neurons above threshold. In conclusion, the recruitment of neurons is avalanche-like, i.e., slow cells, which are more inert to fluctuations are swept along in a later stage of the PS.

Initiation and timing of PSs in the model with cellular AHP

A similar analysis of the generation and temporal occurrence of PSs was performed for networks in which synaptic depression was replaced by cell-intrinsic AHP (Fig. 5A). Slow AHPs generated after a few action potentials are prominent in cortical neurons and contribute to the refractory period during slow cortical oscillations (Sanchez-Vives and McCormick 2000). The synaptic connection strength was varied over a large range of parameter space (see Figs. 2 and 7). We first studied values of the synaptic connection strength close to the minimal value for which spontaneous synchronization can occur (see Fig. 5). This value is approximately 10-fold smaller than for the model with synaptic depression, because there is no use-dependent decrease in the synaptic resources [average (xij) «1 for the model with synaptic depression, and average (xij) = 1 for the model with AHP]. The sum of AHP amplitudes (VAHPtot) was used to quantify the overall refractoriness of the network. A PS generated in this model results in a large increase in the net AHP amplitude that recovers on a time scale comparable to the time constant of recovery of VAHP (Fig. 5B).

Similar to the model with synaptic depression, interspike intervals appear irregular, and the time course of VAHPtot does not allow for the deterministic prediction of the occurrence of the next PS, suggesting a stochastic mechanism for the generation of PSs. However, in contrast to the case of synaptic depression, the time course of VAHPtot after a PS deviates from a monoexponential relaxation at late times (Fig. 5, A and B), showing marked fluctuations on the way to recovery. In particular, some of the fluctuations are so significant that they lead to a considerable reset of VAHPtot without involving the large number of neurons characteristic for a full PS (see e.g., the fluctuation in Fig. 5A between 15 and 20 s). Inspection of the time trace of the sum of AHP amplitudes allows us to attribute the side-peak in the histogram of Fig. 5C (around 4,000 ms) to such fluctuations, while the most likely duration of interspike intervals is between 2 and 3 s (see Fig. 5C). This stochastic element in the approach to recovery is not contained in Eq. 7. It plays a negligible role in the model with synaptic depression (Fig. 3C), which shows a much smoother time course of recovery, and it becomes less important for increasing connection strength in the model with cellular AHP (Fig. 7). The simulation shown in Fig. 5 illustrates that, in the presence of large-scale fluctuations, the dynamics of recovery from refractoriness is subject not only to deterministic but also to stochastic factors, and a description of the collective network activity in terms of Poissonian processes is not valid. As a consequence, the fit of Eq. 7 to the distribution of interspike intervals was poor (χ2/ndf = 10.8; Fig. 5C, determined from 226 PSs occurring during 731 s of simulation).

In further contrast to the case of synaptic depression, PS-triggered averages reveal no temporal ordering of neuronal subgroups in the recruitment process (Fig. 6, A and B). To understand this, we inspected membrane voltage traces for single cells prior and during the generation of a PS (Fig. 6C). In general, firing occurs if the synaptic input RinIsyn(i,t) = ΣjAijyij(t) specified in Eq. 2 is comparable to the distance VthrVi. This firing condition can be achieved either by keeping the difference VthrVi relatively unchanged while increasing the synaptic connection strength Aij. Alternatively, for a time-independent synaptic connection strength, a time-dependent decrease of VthrVi makes firing more likely. The former condition is realized in the model with synaptic depression, the latter in the model with AHP. Thus prior to a PS, predominantly silent neurons are much closer to threshold in the model with AHP (cf. Figs. 4C and 6C). Moreover, the membrane potentials of all cells participating in the ensuing PS show fluctuations smaller, but comparable in size, to their distance from threshold. Thus all these cells can directly respond to fluctuations in background activity with the generation of an action potential. This is in marked contrast to the model with synaptic depression (Fig. 4C), where fluctuations comparable in size to the distance from threshold are limited to the small subgroup of middle cells, thus rendering these cells functionally distinct triggers. Consequently, synchronization emerges here as a simultaneous recruitment of a large, easily excitable population of neurons. This model therefore provides a mechanistically different example for a network capable of initiating synchronized discharges in response to self-generated background fluctuations.

fig. 6.

In the model with AHP, the generation of PSs occurs via simultaneous recruitment of all neurons. A: two examples showing how slow, middle, and fast cells contribute to a PS. B: PS-triggered averages for these three neuronal populations ±100 ms around the peak of the PS (n = 226). Bottom: fraction of neurons from the 3 neuronal subpopulations recruited during a PS, highlighting their nearly simultaneous discharge. Color code is the same as the one used in Fig. 4. C: representative membrane potential traces from neurons from the slow (cells 1, 100, 200, 300) and the middle group (cell 350) during the generation of a PS. Cells 1, 100, 200, 300 are close to threshold and are thus recruited. In contrast, neuron 350 discharges accidentally before the PS and is not available for the PS. Traces are displaced horizontally by 100 ms.

We further note that the high excitability of individual neurons leads to some accidental generation of action potentials during the periods free of PSs (Figs. 1F and 5A). These events may be classified as “mini-avalanches,” which do not develop into full PSs since the network has not recovered sufficiently from its refractory state. Such mini-avalanches recruit relatively few silent neurons (see Fig. 1F). However, they have a tendency to recruit the most recovered ones, thus leading to significant small-scale fluctuations in the sum of AHP amplitudes (Fig. 5A). As a consequence, action potentials generated in individual neurons of the predominantly silent neuronal groups are unreliable indicators of an imminent PS. This unreliability also underlies the variations in the number of cells available for recruitment during a PS, explaining the variable amplitudes of VAHPtot in Fig. 5A. This variability decreases with increasing synaptic connection strength when the tendency to generate action potentials accidentally outside PSs is also diminished.

Comparison of the models

In the simulations presented so far, the synaptic connection strengths were set close to the minimal values for which spontaneous synchronization was possible (see Fig. 7). Since synaptic weights are adapted dynamically in the model with synaptic depression, while they remain constant in the model with AHP, we studied to what extent the timing of PSs in the two models is affected differently by changes in the synaptic connection strength. Both network models result in similar distributions of firing frequencies (Fig. 7), which depend only weakly on the synaptic connection strength and for which approximately two thirds of the neuronal population fire only during PSs. Moreover, for both models, the timing and shape of PS-triggered averages is similar for the case of large (Figs. 7, A and B) and small (Figs. 4 and 6) synaptic connection strength, indicating that the avalanche-like recruitment mechanism within a PS remains essentially unchanged. Thus independent of the connection strength, important global network characteristics such as neuronal frequency distributions and recruitment mechanisms remain comparable. Moreover, the average interspike interval between PSs (Fig. 7, C and D) decreases in both models with increasing synaptic strength, indicating that the strength of background fluctuations is reflected in the timing of PSs. This decrease shows a lower bound related in the one case to the membrane time constant τ = 30 ms and in the other case to τAHP = 600 ms.

The remarkable difference between the two models is that, with increasing connection strength, the model with AHP shows a transition from a stochastic to a clockwise occurrence of PSs, whereas the model with synaptic depression retains a stochastic timing. This is quantified in Fig. 7, C and D, by the deviation of interspike intervals from their average (defined as the width of the distribution of interspike intervals normalized to their mean). For the model with AHP, this width narrows strongly with increasing connection strength (from approximately 30% to <3%), but it remains largely unaffected in the model with synaptic depression (approximately 30%). Thus the model with synaptic depression maintains a fluctuation-dominated behavior for large connection strength. Also the distribution of interspike intervals remains of the stochastic form given in Eq. 8 (data not shown).

The different sensitivity of the timing of PSs to the value of the connection strength indicates that, via different refractory mechanisms, the overall excitability of the network is reflected differently in the temporal statistics of synchronized discharges.


The main focus of this study was on the generation and timing of spontaneous neuronal synchronization in recurrently connected network models. In the absence of a pacemaking drive, fluctuations in spontaneous activity serve as a trigger for sharply synchronizing large fractions of the network. This synchronization process makes use of recruitable subpopulations of less active cells, which as amplifiers of spontaneous fluctuations, spread activity in an avalanche-like manner that can be sequential or simultaneous. Synchronization of discharge is independent of the precise mechanism of neuronal refractoriness, but the mechanistic implementation of refractoriness governs the temporal distribution of synchronized events.

The ability of recurrently connected networks to generate temporally patterned synchronization has been emphasized repeatedly in model studies. In firing-rate models, which follow the time course of average network quantities, it has been demonstrated that the interplay between excitatory connectivity and cellular or synaptic refractoriness can account for episodic alteration between states of high and low activity (Senn et al. 1996; Tabak et al. 2000). Similar findings were reported from model studies, which follow simultaneously the time course of hundreds of interconnected neurons (Butts et al. 1999; Timofeev et al. 2000; Tsodyks et al. 2000). In addition, this latter approach has revealed that gradual differences within intrinsic and synaptic properties of neuronal subgroups can affect significantly their function in the generation of PSs (Tsodyks et al. 2000). Despite this recent progress, however, the question of how synchronized activity arises dynamically from single cell properties has remained open (Feller 1999; Zhang and Poo 2001).

This study does not aim to reproduce a realistic network to address this question. Instead, we established for two relatively simple network models how the interplay between deterministic factors, originating from the biophysics of single neurons, and from stochastic factors initiates and times synchronized network activity. While the network models studied here are strictly deterministic, stochastic behavior emerges as a consequence of a chaotic dynamical evolution in which memory about the initial network condition or the occurrence of the last PS is quickly lost. This makes the flat asynchronous background activity of fast cells functionally equivalent to an external noise source. The overall network refractoriness, which we quantified as the sum of synaptic reserve in the network (Fig. 3A, bottom panel) or as the sum of AHP amplitudes (Fig. 5A, bottom panel), may be regarded as the only long-term memory which the network keeps about its past. With the deterministic recovery from refractoriness, the initiation probability of avalanches increases gradually, and thus a stochastic spread of interspike intervals can emerge from a fully deterministic system. What matters for the occurrence of PSs is that the probability for the flat stochastic background activity to excite neurons in the predominantly silent subpopulation is reduced temporarily after a PS. This can be achieved either by leaving their membrane potential unmodified but weakening their excitability via the mechanism of synaptic refractoriness. Alternatively, the excitability of predominantly silent neurons can be reduced via a cell-intrinsic mechanism such as a slowly decaying AHP, while synaptic connections remain unchanged. We emphasize that in both implementations of refractoriness, the change in excitability is deterministic while the actual excitation process amplifies a stochastic fluctuation. However, the relative importance of these deterministic and stochastic factors is very different, resulting in periodicity in the model with AHP but stochastic behavior in the model with synaptic depression (Fig. 7).

Initiation and timing of patterned electrical activity, and consequently of fluctuations of intracellular second messengers, are important for determining developmental processes. Thus the temporal properties of episodic synchronization are crucial for appropriate patterning of connections (Feller 1999; Stellwagen and Shatz 2002; Wong 1999) and for controlling growth cone properties and expression of neurotransmitters (Spitzer 2002; Zhang and Poo 2001). On the other hand, developing neural networks show a striking robustness in the generation of temporal patterns of activity, being capable of synchronization via different mechanisms at a single developmental age, or independently of developmental changes in neurotransmitter release (Feller 1999; Wong 1999). Our model suggests a dynamical basis for this versatility by showing that, while no highly precise neural circuitry is required for producing synchronization, the regulation of intrinsic and synaptic properties can control initiation and timing over a broad range of connection strengths.

Our study provides two models for spontaneous rhythmogenesis emerging out of a minimal network structure. It illustrates in detail how seeds of synchronization and functionally different neuronal subpopulations can arise dynamically from a uniform network structure with a natural continuous spread of single-cell properties but without functionally specialized neurons. It is plausible that developing neuronal networks make use of this mechanism of diversification, both by structuring their activity in response to their self-generated noisy background and by using genetically and architecturally preestablished gradients to attribute functional specification (Feller 1999). Indeed, in the developing retina, there is a continuum in the degree of spontaneous activity of single cells and of their participation in spontaneous waves, suggesting that there exists a natural spread of neuronal properties in developing networks (Butts et al. 1999; Feller et al. 1996).

Despite its structural and physiological simplicity, this study may capture features typical for spontaneous synchronization that also hold in adult systems devoid of an identified pacemaker. For example, the “slow oscillation” (approximately 1 Hz), a cortically generated sleep rhythm (Steriade et al. 1993b), can be generated in ferret neocortical slices in vitro upon slight enhancement of neuronal excitability (Sanchez-Vives and McCormick 2000) and appears to result from a synchronization of neuronal discharge across cortical layers. Initiation is triggered by recruitment of the subpopulation of silent supragranular neurons following fluctuating action potential discharge in infragranular neurons, while timing is correlated with a slowly decaying AHP. It is an intriguing question, emerging out of this study, whether the temporal statistics as well as the dynamical origin of this cortical rhythm can be understood in terms of the interplay between stochastic properties and refractory mechanisms.


This work was supported by a Swiss National Science Foundation Grant 31-61434.00 to A. Lüthi.


We thank Profs. Beat H. Gähwiler and Urs Gerber for constructive comments and critical reading of earlier versions of the manuscript.


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